Best Linear Time-Varying Approximation of a General Class of Nonlinear Time-Varying Systems

Abstract

This article presents a method for estimating a linear time-varying approximation of a general class of nonlinear time-varying (NLTV) systems. It starts from noisy measurements of the response of the NLTV system to a special class of periodic excitation signals. These measurements are subject to measurement noise, process noise, and a trend. The proposed method is a two-step procedure. First, the disturbing noise variance is quantified. Next, using this knowledge, the linear time-varying dynamics are estimated together with the NLTV distortions. The latter are split into even and odd contributions. As a result, the signal-to-nonlinear-distortion ratio is quantified. It allows one to decide whether or not a linear approximation is justifiable for the application at hand. The two-step algorithm is fully automatic in the sense that the user only has to choose upper bounds on the number of basis functions used for modeling the response signal. The obtained linear time-varying approximation is the best in the sense that the difference between the actual nonlinear response and the response predicted by the linear approximation is uncorrelated with the input. Therefore, it is called the best linear time-varying approximation (BLTVA). Finally, the theory is validated on a simulation example and illustrated on two measurement examples: the crystallographic pitting corrosion of aluminum and copper electrorefining.